#72279211

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#158797 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#15480⋊C₃
commutator, cosocle	C₂³×C₄²	C₂²×C₃
Frattini, socle	C₂⁵	C₂²×A₄
Fitting	#158797	C₃
prime radical	C₂×C₂²⋊(C₄²⋊C₃)	C₂²
hypocenter	C₂²×C₄²	C₃×Q₈
LCS	G ▷ C₂³×C₄² ▷ C₂²×C₄²
UCS	C₁ ◁ C₂

1074 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^2,3,2^7|a^2=d,b^2=d,e^2=j,f^2=i,b^a=bd,c^a=cij,c^b=ci,e^a=eg,e^b=eh,e^c=ef,f^a=fghij,f^b=fgj,f^c=ei,g^c=hij,h^c=ghj,i^c=j,j^c=ij

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72299775
maximal quotient of	#72299986

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 3, F.5*F.9);;
gap> SetConjugate(coll, 7, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.10);;
gap> SetConjugate(coll, 10, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);